Modeling Ionic Liquids Using a Systematic All-Atom Force Field

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Modeling Ionic Liquids Using a Systematic All-Atom Force Field

Jose´ N. Canongia Lopes*

Centro de Quı´mica Estrutural, Instituto Superior Te´cnico, AV. RoVisco Pais, 1049-001 Lisboa, Portugal

Johnny Deschamps and Agı´lio A. H. Pa´dua

Laboratoire de Thermodynamique. des Solutions et des Polyme`res, UniVersite´ Blaise Pascal - 24 AV. des Landes, 63177 Aubie`re, France

ReceiVed: July 29, 2003; In Final Form: October 3, 2003

A new force field for the molecular modeling of ionic liquids of the dialkylimidazolium cation family was constructed. The model is based on the OPLS-AA/AMBER framework. Ab initio calculations were performed to obtain several terms in the force field not yet defined in the literature. These include torsion energy profiles and distributions of atomic charges that blend smoothly with the OPLS-AA specification for alkyl chains.

Validation was carried out by comparing simulated and experimental data on fourteen different salts, comprising three types of anion and five lengths of alkyl chain, in both the crystalline and liquid phases. The present model can be regarded as a step toward a general force field for ionic liquids of the imidazolium cation family that was built in a systematic way, is easily integrated with OPLS-AA/AMBER, and is transferable between different combinations of cation-anion.

1. Introduction

Ionic liquids (ILs) are a class of organic salts with unusually low melting temperatures that attracted much attention from the scientific community in recent years. Apart from being liquid around room temperature, ionic liquids exhibit a negligible vapor pressure and interesting solvation or coordination properties that depend on the characteristics of the cation and anion. Their status as “green” or “designer” solvents explains the large amount of studies concerning their possible industrial use as reaction or extraction media.¹Unfortunately, their thermophysical charac- terization is by no means extensive. The amount and scope of molecular modeling studies are also limited.²^-⁷

In this article we introduce a force field to address three main issues. (i) Internal consistency. Many of the existing IL models borrow parameters from different, and not always compatible, sources: it is common to see parametrizations of the cation and the anion using information from different force fields. In this work ab initio calculations are used extensively to provide essential data for the development of an internally consistent force field. This includes molecular geometry optimization and the description of electron density using extended basis sets, leading to the evaluation of force field parameters such as torsion energy profiles and point charges on the interaction centers. (ii) Transferability. The parametrization of IL models should not be too specific because one wants to deal with families of similar compounds, generally the cations, that can be combined with different counterions. The proposed IL force field will concen- trate on the parametrization of parts of the molecules that are common to an entire family of cations and adopts a strategy to add specific substituents. (iii) Compatibility. The model will build on the OPLS-AA force field.⁸ This means that any molecule or residue already defined in the OPLS-AA database can be combined with the proposed force field. The choice of the OPLS-AA force field as a framework also benefits from its

articulation with structural parameters from the even larger AMBER force field.⁹

The most difficult point at this stage is the validation of any given IL model due to the scarce and sometimes inaccurate experimental data. Obviously, the present effort to achieve consistency does not necessarily guarantee accuracy. Despite the limited experimental data, the proposed force field will be tested against relevant experimental and simulation results from the literature.

The most extensively studied and modeled IL family has been that of the 1-alkyl-3-methylimidazolium cation (cf. Figure 1 for the atomic nomenclature adopted throughout this paper) with different counterions, PF6-

, BF4-

, Cl^-, etc. Because the Figure 1. Nomenclature of the 1-alkyl-3-methylimidazolium cations.

AMBER-like notation was used on the ring atoms.⁹C1refers to the first carbon atom in any side chain, CEis the second carbon atom in the ethyl side chain of 1-ethyl-3-methylimidazolium, C2is the second atom in alkyl side chains with more than 2 carbon atoms, CSrefers to any secondary carbon of the alkyl side chain that is removed at least 2 bonds from the ring, and CTis the terminal carbon atom of longer than ethyl side chains. HA, H1, and HCrefer to hydrogen atoms attached to the aromatic ring, to the first carbon atom on the side chains, and to carbon atoms on the side chain further removed from the ring, respectively. Abbreviations used for 1-alkyl-3-methylimidazolium cations are mim⁺)methylimidazolium, with prefixes a)alkyl, m) methyl, e)ethyl, b)butyl, h)hexyl, dd)dodecyl.

2038 J. Phys. Chem. B 2004, 108, 2038-2047

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validation of the molecular model is always a primary concern, the development of the present IL model will focus on the family of this particular cation and some of the most common anions.

2. Force Field Development

The parametrization of a force field and its validation using simulation techniques requires the knowledge of four sets of molecular data: (i) molecular geometry; (ii) intramolecular force field, applicable in the case of nonrigid molecules; (iii) crystalline geometry, applicable in the case of simulations of ordered phases; and (iv) intermolecular potential. The first two sets comprise internal molecular parameters that can be esti- mated from quantum mechanical calculations and/or gas-phase spectroscopic data whereas the last two sets include external or intermolecular parameters to be inferred from auxiliary diffraction and/or bulk thermodynamic data. Sets ii and iv contain the so-called force field parameters.

The currently available IL force field models used different approximations to build each set of parameters. The differences and similarities are summarized in Table 1 along with the model developed in this paper.

Molecular Geometry. The molecular geometry parameters are generally obtained from low-level ab initio calculations using a simple basis set describing a single molecule (cf. Results section below). All the models under discussion used a similar method to estimate the parameters describing the imidazolium ring (bond lengths, angles, and dihedrals) and the published results are comparable with experimental data obtained by diffraction studies, as shown in Table 2. Two conclusions can be drawn directly from the table: (i) the ring geometry is not strongly affected by the environment of the imidazolium molecule because the ring geometries in two completely

different crystals, [emim][VOCl4] and [ddmim][PF6],^21,22 are comparable between them and similar to the ab initio values for the isolated imidazolium cation, and (ii) the distortion of the ring caused by different alkyl substituents is so small that the use of a symmetrical ring geometry as in a mmim⁺cation represents a good approximation.

In fact, the geometry of the imidazolium cation ring can be qualitatively rationalized if one compares it with the isoelec- tronic structures of the cylclopentadienyl anion, pyrrole, and imidazole, as shown in Figure 2. Whereas the cylclopentadienyl anion is a regular pentagon with average C-C distances around 139 pm,¹⁰the imidazolium ring is somewhat smaller and slightly deformed due to the shorter N-C bonds and to the presence of nonequivalent contributors to the resonance hybrid. In the imidazolium cation the NA-CWdistances (around 138 pm) are comparable to the N-C distance in pyrrole (137.8 pm) and correspond to a single bond; the CW-CWdistances (134.1 pm) are characteristic of C-C double bonds (134 pm); and the short NA-CRdistances (131-132 pm) represent a “one-and-an-half”

N-C bond.

The internal angles on the imidazolium ring do not deviate significantly from the regular pentagon angle of 108°: the CR

tip is somewhat less acute (109.8°), and the CWtips compensate that distortion by being more acute (107.1°). The other small but noticeable distortion is on the angles to the atoms connected at NAand CW. For instance, the angle between the alkyl side chains and the CRatom (C-NA-CRangle) are always slightly larger than the angle to the CWatom (C-NA-CWangle).

All dihedrals on the imidazolium ring calculated from ab initio methods are approximately either 0°or 180° so the ring and the atoms directly connected to it are on the same plane. This is not always true in the case of experimental diffraction data TABLE 1: Available and Proposed Force-Field Models for Ionic Liquids

model (authors)

HPLB [ref 2] ABS [refs 3,5] SBM [ref 4] MM [ref 7] MSB [ref 6] CLDP [this work]

Ions Modeled

cations mmim⁺, emim⁺ emim⁺, bmim⁺ bmim⁺ bmim⁺ bmim⁺ amim⁺, (C1-C12)

anions Cl^-, PF6- AlCl4-, BF4- PF6- PF6- PF6- PF6-, Cl^-, NO3-

Internal Geometry optimization

level of theory HF UHF RHF B3LYP HF RHF

basis set 6-31G(d,p) 6-31G(d) 6-31G(d) 6-31G(d) 6-31G(d,p) 6-31G(d)

electron density

level of theory MP2 UHF RHF B3LYP HF MP2

basis set 6-31G(d,p) 6-31G(d) 6-31G(d) 6-311+G(d) 6-31G(d,p) cc-pVTZ(-f)

partial charges DMA RESP CHelpG CHelpG ESP CHelpG

Internal Force Field

cation ring^a rigid AMBER rigid CHARMM OPLS-AA OPLS-AA

cation alkyl chain free dihedrals AMBER OPLS-UA,

OPLS-AA dihedrals

CHARMM OPLS-AA OPLS-AA,

estd by QM/MM

anion rigid estd by QM/MM one center estd by QM rigid rigid

External Force Field

parametrization Exp6+q LJ+q LJ+q LJ+q LJ+q LJ+q

cations Williams AMBER OPLS-UA CHARMM OPLS-AA OPLS-AA

anions Williams,

MMFF94

AMBER, DREIDING

SF6data CHARMM OPLS-AA OPLS-AA,

AMBER

MD method MD method MC method MD method MD method MD method

initial cell geometry CCDB, private comm. (crystal), random (fluid)

grid with randomly positioned ions (fluid)

random (fluid) random (fluid) expanded lattice, random (fluid)

CCDB (crystal), expanded lattice, random (fluid)

production length 5×100 ps 100 ps 2×5000 cycl 4 ns 20×50 ps 150-350 ps

Validation

internal analysis RDF, energy RDF, energy RDF, energy RDF, energy RDF energy

experimental data crystal structures, density

neutron scattering, density, self-difusion

density,R_p,βT IR data, density, R_p,βT,self-difusion

self-difusion crystal structures, density

aThe five atoms in the ring plus the five atoms directly attached to them.

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where the crystal geometry can impose slight out-of-the-plane distortions on the atoms connected to the ring.²²

The geometry of the alkyl side chains (C-C and C-H bond lengths, H-C-H, C-C-C, and C-C-H angles) are similar to those found in normal aliphatic chains. The carbon in the side chain connected to the NAatom is counted as part of the ring and not as a proper member of the side chain.

To establish the molecular geometry of the PF6-

and NO3-

anions (taken as octahedral and planar triangular structures, respectively), it is sufficient to know the P-F and N-O interatomic distances. These were taken from ab initio data^24,26 and are also presented in Table 2. The corresponding distances obtained from X-ray diffraction studies in selected IL crystals compare favorably with the ab initio results.

Intramolecular Force Field. Many bond, angle, and dihedral parameters used in this work were taken from or based on the OPLS-AA and AMBER force fields, which are largely compatible with respect to the intramolecular terms. Whenever we found significant differences between our ab initio geometries and OPLS-AA or AMBER parameters, notably in equilibrium distances and angles, we proposed new values, reported in Table 3. We did not modify force constants from what is specified in

AMBER and OPLS-AA, because these features have a relatively small effect on thermodynamic properties.

The force field parameters were developed using a procedure similar to previous studies concerning heterocyclic five- membered rings.²⁸Because the OPLS-AA/AMBER database does not contain information about the imidazolium cation itself, the bond lengths of the ring were chosen to reproduce the ab initio geometry of the isolated cations (see previous section) whereas the relevant stretching and bending force constants of the ring were inferred from the values published for similar heterocyclic compounds such as imidazole, pyrrole, and purines.^9,28,29The main departure from the OPLS-AA force field at this point regards our use of constrained bond lengths for all C-H stretching modes present in the cations (cf. Results section below).

Margulis et al.⁶also used OPLS-AA/AMBER parameters to define the internal force field of the 1-butyl-3-methylimidazolium cation but chose to model the ring directly with imidazole parameters (neutral and asymmetric ring). The same applies to the work of de Andrade et al.:⁵although the CHARMM force field contains parametric data concerning the protonated histidine ring, the authors chose parameters that can be understood TABLE 2: Molecular Geometry of the 1-Alkyl-3-methyl-imidazolium Cation Family and the PF6-and NO3-Anions^a

emim⁺XRb[ref 21] ddmim⁺XRc[ref 22] emim⁺AI[ref 3] emim⁺AId[ref 4] amim⁺AIe[this work]

Bonds (Å)^f

NA-CR 1.311(4) 1.322(3) 1.326(3) 1.314 1.315

1.311(4) 1.315

NA-CW 1.357(5) 1.373(3) 1.378 1.378

1.360(6) 1.374(3) 1.378

CW-CW 1.334(8) 1.334(8) 1.342 1.341

NA-C1 1.452(4) 1.468(3) 1.466 1.466

1.468(4) 1.477(3) 1.478

C1-CE 1.500(9) 1.520

Angles (deg)^f

NA-CR-NA 109.6(3) 109.8 109.9 109.8

NA-CW-CW 107.1(3) 108.1 107.0 107.1

107.6(4) 108.0 107.2

CW-NA-CR 108.0(3) 106.9 107.9 108.0

107.6(2) 106.8 108.0

CW-NA-C1 125.9(3) 125.6 125.6

125.2(3) 125.9

CR-NA-C1 126.5(3) 126.4 126.3

125.4(3) 126.1

PF6-

XRg[ref 23] PF6-

XRb[ref 22] PF6-

AI[ref 24] NO3-

XRh[ref 25] NO3-

AI[ref 26]

Bonds (Å)

P-F 1.566 1.596 1.560

N-O 1.220 1.250

aComparison between experimental crystal X-ray (XR) geometries and single-molecule ab initio (AI) calculations. The results in bold are those used in the parametrization of the present force field ^b[emim][VOCL4] crystal.^c[ddmim][PF6] crystal.^dCalculated from molecular coordinates given in the reference.^eAverage values, valid for an imidazolium ring with any alkyl susbstituent.^fDouble entries refer to bond lengths and angles on the alkyl and methyl side of the imidazoilum ring, respectively.^g[emim][PF6] crystal.^h[emim][NO3] crystal.

Figure 2. Geometry of the imidazolium cation (im⁺), cyclopentadienyl anion (cp^-), pyrrole (py), and imidazole (im°) rings. All bond distances are in picometers. The lines inside each ring represent the delocalized π-electrons in the corresponding aromatic systems. The fourπ-electrons predominantly shared between three atoms in im⁺and the sixπ-electrons shared between five atoms in cp^-are shown as curved lines. The lone pair of electrons in one of the nitrogen atoms in im°is also shown.

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TABLE 3: 1-Alkyl-3-methylimidazolum Force-Field Parameters^a

atoms q (e) σ(Å) (kJ mol^-¹) C1 C2 C4 C6 C12 C18 source

C1 -0.17 3.50 0.27614 2 2 2 2 2 2 this work (q),

OPLS (butane)⁸

C2 0.01 3.50 0.27614 1 1 1 1 this work (q),

OPLS (butane)⁸

CE -0.05 3.50 0.27614 1 this work (q),

OPLS (butane)⁸

CR -0.11 3.55 0.29288 1 1 1 1 1 1 this work (q),

OPLS (HIP^d)²⁷

CS -0.12 3.50 0.27614 1 3 9 15 OPLS (butane)⁸

CT -0.18 3.50 0.27614 1 1 1 1 OPLS (butane)⁸

CW -0.13 3.55 0.29288 2 2 2 2 2 2 this work (q),

OPLS (HIP^d)²⁷

HA 0.21 2.42 0.12552 3 3 3 3 3 3 this work (q),

OPLS (HIP^d)²⁷

HC 0.06 2.50 0.12552 3 7 11 23 35 OPLS (butane)⁸

H1 0.13 2.50 0.12552 6 5 5 5 5 5 this work (q),

OPLS (butane)⁸

NA 0.15 3.25 0.71128 2 2 2 2 2 2 this work (q),

OPLS (HIP^c)²⁷

bonds req(Å) Kr(kJ mol^-1Å^-2) C1 C2 C4 C6 C12 C18 source

CR/W-HA 1.08 (constrained) 3 3 3 3 3 3 OPLS (imidazole)²⁸

C*-H*^b 1.09 (constrained) 6 8 12 16 28 40 OPLS (butane)⁸

CR-NA 1.315 1996 2 2 2 2 2 2 this work (r),

OPLS (imidazole)²⁸

CW-NA 1.378 1787 2 2 2 2 2 2 this work (r),

CW-CW 1.341 2176 1 1 1 1 1 1 this work (r),

NA-C1 1.466 1410 2 2 2 2 2 2 this work (r),

AMBER(N*-CT)^9,29

C*-C*^b 1.529 1121 1 3 5 11 17 OPLS (butane)⁸

angles θeq(deg) Kθ(kJ mol^-1rad^-2) C1 C2 C4 C6 C12 C18 source

CW-NA-CR 108.0 292.6 2 2 2 2 2 2 this work (θ),

CW-NA-C1 125.6 292.6 2 2 2 2 2 2 this work (θ),

CR-NA-C1 126.4 292.6 2 2 2 2 2 2 this work (θ),

NA-CR-HA 125.1 146.3 2 2 2 2 2 2 this work (θ),

NA-CR-NA 109.8 292.6 1 1 1 1 1 1 this work (θ),

NA-CW-CW 107.1 292.6 2 2 2 2 2 2 this work (θ),

NA-CW-HA 122.0 146.3 2 2 2 2 2 2 this work (θ),

CW-CW-HA 130.9 146.3 2 2 2 2 2 2 this work (θ),

NA/C*-C*-H*^b 110.7 313.2 6 10 18 26 50 74 OPLS (butane)⁸

NA/C*-C*-C*^b 112.7 418.4 4 3 5 11 17 OPLS (butane)⁸

H*-C*-H* 107.8 276.1 6 4 9 11 17 23 OPLS (butane)⁸

dihedrals V1(kJ mol^-1) V2(kJ mol^-1) V3(kJ mol^-1) C1 C2 C4 C6 C12 C18 source

X-NA-CR-X 0 19.46 0 8 8 8 8 8 8 AMBER⁹

X-CW-CW-X 0 44.98 0 4 4 4 4 4 4 AMBER (X-CC-CW-X)⁹

X-NA-CW-X 0 12.55 0 8 8 8 8 8 8 AMBER⁹

CW-NA-C1-H1 0 0 0.55 6 5 5 5 5 5 this work

CR-NA-C1-H1 0 0 0 6 5 5 5 5 5 this work

CW-NA-C1-C2/E -5.76 4.43 0.877 1 1 1 1 1 this work

CR-NA-C1-C2/E -3.23 0 0 1 1 1 1 1 this work

NA-C1-C2-CS/T 0.738 -0.681 1.02 1 1 1 1 this work

NA-C1-C2/E-HC 0 0 0 3 2 2 2 2 this work

C*-C*-C*-H* 0 0 1.531 9 17 39 63 OPLS (butane)⁸

H*-C*-C*-H* 0 0 1.331 6 14 22 48 72 OPLS (butane)⁸

C*-C*-C*-C* 0.728 -0.657 1.167 1 3 9 15 OPLS (butane)⁸

X-NA-X-X^c 0 8.37 0 4 4 4 4 4 4 AMBER (X-N*-X-X)⁹

X-CW/R-X-X^c 0 9.2 0 6 6 6 6 6 6 AMBER^9,29

aC1 to C18 represent the number of occurrences of each term in a given amim cation designated by the number of carbons in the longer side chain.^bNo difference found in the equilibrium angles and force constants of aliphatic carbons connected either to carbon or nitrogen. C* represents a generic aliphatic carbon, C1, C2, CE, CS, or CT. H* represents either H1or HC.^cImproper dihedral.^dHIP is the protonated histidine cation.

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as averages of the two (asymmetrical) halves of the neutral histidine ring. Unlike the imidazolium ring, the imidazole ring residue is not symmetrical and the distortions of the structure around the carbon connected to the two (nonequivalent) nitrogen atoms are noticeable (cf. distances around CRin imidazole in Figure 2). The use of imidazole bond distances and angles to define an imidazolium cation can lead to an inaccurate parametrization.

In the present work, the stretching and bending force constants related to the bonds between the ring and the alkyl side chains were taken from AMBER parameters used to model similar bonds in purines (adenine and guanine included in nucleic acids).⁹It should be noted that the OPLS-AA force field adopts the same force constants when modeling the bonds and angles involving the N-C pair in amines and amides.²⁷The corresponding values, both force constants and distances, for the alkyl side chains were taken from the OPLS-AA publication covering aliphatic chains.⁸The parameters defining the torsion energy profiles in the imidazolium ring were taken from the AMBER force field concerning heterocyclic aromatic molecules. These include the improper dihedrals needed to maintain the ring and the atoms directly attached to it in a planar geometry. The OPLS-AA specification of five-membered heterocyclic aromatic rings²⁸constrained the ring and the atoms directly connected to it to be planar, which means that no dihedrals involving those atoms needed to be defined. Nevertheless, the authors of OPLS- AA recommend the use of AMBER parameters for cases where there are no ad-hoc constraints. The torsion profiles along the aliphatic side chain were obtained from the published OPLS- AA results.⁸ The missing parameters corresponding to the dihedrals between atoms belonging to the imidazolium ring and those in the alkyl side chain were defined in this work with the aid of ab initio calculation (cf. Results).

As regards the anions simulated in this work, they were modeled as rigid molecules. No intramolecular force field parametrization is therefore needed, apart from the interatomic distances given in the previous section. Of course one might object that, because the cations were modeled as flexible molecules, we should have used a similar approach when dealing with the anions. Two facts justify our present choice: (i) The anions under discussion are not articulated molecules (there are no proper dihedral angles involved). This means that using fixed bond lengths and angles will not hinder any internal movement of the molecule. And (ii) there is a large uncertainty associated with the parametrization of the external force field of these ions;

cf. Intermolecular Potential section below. In our opinion the use of a more complete internal force field is not relevant at this stage, before these (and other) anions have a better defined set of external parameters.

Crystal Geometry. The ionic liquids based on the 1-alkyl- 3-methylimidazolium cation family exhibit higher melting-point temperatures for cations with either very short or long alkyl side chain lengths: mmim⁺and emim⁺with Cl^-, PF6-

, or NO3-

and also [ddmim][PF6] have melting points above 273 K. The crystalline structures of these salts are documented at the Cambridge Crystallographic Database and can be used to validate the force field parametrization. Such a procedure was used by Hanke et al.²to test their model describing the above- mentioned mmim⁺and emim⁺salts. In the present work the prediction of the crystalline structure using the proposed force field is extended to salts containing the ddmim⁺cation and the nitrate anion. The crystallographic data needed to establish the structure of each salt will be given along with the simulation results (cf. Results).

Intermolecular Potential. The OPLS-AA/AMBER external force fields both comprise repulsion-dispersion terms parametrized by a 12-6 Lennard Jones potential function and an electrostatic term represented by partial charges located at each interaction site of the molecule.

In the case of the imidazolium cations, the Lennard-Jones parameters for each type of atom were taken from the OPLS- AA parametrization of heterocyclic aromatic rings²⁸or aliphatic compounds⁸ (Table 3). The interactions between atoms of different type were parametrized using the Lorentz-Berthelot mixing rules (arithmetic and geometric mean rules forσand, respectively). The partial charges on each atom were calculated from the electron density obtained by ab initio calculation, using an electrostatic surface potential methodology (cf. Results).

The partial charges and Lennard-Jones parameters in the PF6-

anion were taken directly from the OPLS-AA force field.²⁴The NO3-

ion was modeled by four Lennard-Jones centers whose parameters were determined by fitting to the Born-Mayer potential used to describe dispersive interactions in molten salts.¹¹The partial charges had been calculated ab initio. The parameters for Cl^-were obtained in a similar way, fitting to the Born-Mayer potentials developed for crystalline and molten salts.¹²At this point it is important to stress that the resulting Lennard-Jones parameters for Cl^-are quite different from those usually employed for chlorine atoms bonded to organic molecules or aqueous Cl^-. The differences are apparent if we compare the interaction diameter,σ, in the three situations: 3.77 Å in molten salts/ionic crystals,¹² around 3.5 Å in organic chlorine atoms,^2,8,9and around 4.4 Å in aqueous solution.⁹All intermolecular parameters concerning the anions presented in this work are compiled in Table 4.

3. Results

Ab Initio Calculations of Torsion Energy Profiles. New torsion potentials corresponding to dihedral angles between the imidazolium ring and the alkyl side chains were developed following a procedure similar to the one adopted for the OPLS- AA force field.^8,13They were obtained from relaxed potential energy scans at a series of values for the dihedral angles of interest, which were held fixed while the remaining internal coordinates of the molecule were allowed to relax to energy minima.

Quantum chemical calculations were performed using Gauss- ian 98¹⁴at the MP2/cc-pVTZ(-f)//HF/6-31G(d) level of theory, thus using the same basis set as in the recent OPLS-AA model for perfluoroalkanes.¹³The cc-pVTZ(-f) basis set¹⁵ was used for single-point energy calculations in geometries optimized at the HF/6-31G(d) level, as is current practice in the development of force field parameters for molecular simulation.^9,16We used frozen-core MP2 calculations instead of the local MP2 method used by Watkins and Jorgensen.¹³The differences between the two methods are small.¹⁷For the C and N atoms the cc-pVTZ- (-f) basis set is created by removing the f functions from the definition of the triple-ζcc-pVTZ basis set of Dunning.¹⁵The combination of the levels of theory and basis sets used here TABLE 4: Lennard-Jones Parameters and Atomic Charges for Selected IL Anions

atoms q (e) σ(Å) (kJ mol^-1) source P, PF6- 1.34 3.7400 0.8368 OPLS²⁴ F, PF6- -0.39 3.1181 0.2552 OPLS²⁴ N, NO3-

0.95 3.0600 0.3380 fit to Born-Mayer¹¹+qAI11

O, NO3- -0.65 2.7700 0.6100 fit to Born-Mayer¹¹+qAI11

Cl, Cl^- -1.00 3.7700 0.6200 fit to Born-Mayer¹²

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has been tested on a large collection of molecules (Halgren test) and was demonstrated to yield accurate conformational energet- ics.¹⁸

The torsion potential energy profiles depending on dihedral angles associated with each chemical bond of interest were described, for the purpose of molecular simulation, by cosine series of the form adopted in OPLS-AA.⁸ The parameters developed in this work correspond to the dihedral angles CR- NA-C1-H1 and CW-NA-C1-H1 (fitted to torsion energy profiles for mmim⁺), CR-NA-C1-C2/E, CW-NA-C1-C2/E, and NA-C1-C2/E-HC(fitted to torsion energy profiles for emim⁺), and NA-C1-C2-C3 (fitted to torsion energy profiles for bmim⁺). This building up method ensures compatibility of the different dihedral functions and allows the description of a homologous series of molecules.

The calculation of the coefficients of the cosine series is not possible by direct adjustment to the ab initio energy profiles.

This complication is due to the presence of nonbonded interactions in the force field that must be accounted for. In fact, in the specification of the OPLS-AA force field, sites within the same molecule separated by more than three bonds interact by Lennard-Jones and electrostatic potentials scaled by a factor of 0.5.⁸Sometimes these nonbonded interactions contribute with a large part of the torsion energy. This means that in the force field, the dihedral functions need not account for the whole torsion energy profiles, because there are contributions from

“steric” effects due to the nonbonded interactions.

The values of the coefficients in the cosine series were found by a fitting procedure requiring structure optimizations with the force field models, performed by molecular simulation of isolated molecules. These MD calculations are done in two steps.¹⁷ First, the geometry of each conformer, for a given constrained value of the dihedral angle, was optimized using molecular simulation with the coefficients of the dihedral function being adjusted set to zero; i.e., the contribution to the torsion energy from that particular dihedral angle came only from nonbonded interactions. After a full scan of the dihedral angle under study is performed, the role of the cosine series will be to correct the torsion profile thus obtained to match the one that resulted from the ab initio calculations described above.

Therefore, the dihedral coefficients were adjusted, by least squares minimization, to the differences between the ab initio energy profiles and those calculated using this incomplete force field.

The simulation procedure employed to optimize the geometries consisted of a series of molecular dynamics quench runs on an isolated molecule, during which 10 000 time steps of 0.5 fs were simulated at each of the following temperatures: 10, 1, 0.1, and 0 K. The dihedrals being scanned were constrained at the desired angles by the addition of very steep harmonic terms where necessary.¹⁷In all cases the range of interest for a given dihedral angle was scanned at 10°intervals, to mimic the set of ab initio calculations. Computer simulations were carried out using the DL_POLY package.¹⁹ The energy of the isolated molecule in its optimized configuration, in which all other torsions, as well as all bonds and all valence angles, where allowed to relax toward a conformational energy minimum, corresponds to one point in the torsion energy profile, for the constrained value of the dihedral being scanned. This procedure is not identical, in terms of the techniques or software tools used, to those employed in the development of OPLS-AA but was validated in a previous work.¹⁷

Once in possession of the adjusted coefficients of the cosine series, the procedure above is repeated, but now with the full

dihedral functions, enabling us to verify that the complete force field actually yields the same torsion energy profile as the ab initio calculations, as seen in Figure 3. We prefer to employ this procedure of relaxed scans using the force field rather than imposing the equilibrium conformations obtained in the ab initio geometry optimizations. Our choice avoids straining the intramolecular terms such as bond lengths, valence angles, and remaining dihedrals, which might not be all perfect matches between the force field and the quantum results. The torsion energy profiles that we develop arise from structure optimizations within the force field itself, using molecular simulation, and this situation corresponds closely to the final use intended for the model, which is the simulation of fluid phases.

The parameters obtained in the present study are collected in Table 3, and a plot of some of the torsion energy profiles and of the representation of the ab initio energies by the force field, is given in Figure 3. The present numerical coefficients were calculated specifically for the imidazolium cation family, using the building up procedure explained above, whenever parameters were not available in OPLS-AA or AMBER. A comparison between the values we obtained here and those of other force fields for imidazolium cations follows. For the dihedral CW-NA-C1-H1, Morrow and Maginn⁷(MM) and de Andrade, Bo¨es, and Stassen^3,5(ABS) set all coefficients to zero, whereas Margulis, Stern, and Berne⁶(MSB) give a value of V3

)2.34 kJ mol^-¹, which is much larger than ours (for nitrogen atoms, instead of type NA, these authors seem to employ parameters for nitrogen atoms of the type NB, which in AMBER notation are nitrogens in five-membered rings with a lone pair).

For the dihedral CW-NA-C1-H1, MM use a nonzero V2

coefficient, ABS set all to zero, and MSB give a value for V3

larger than ours. The rotational barrier associated with the methyl groups in mmim⁺, obtained both by ab initio means and with our parameters, is shown in Figure 3.

For the dihedral CW-NA-C1-C2, MM give a positive V1

coefficient, ABS set this term to zero (according to the AMBER values for generic X-N*-CT-X), and MSB have negative V1and V2coefficients. For the closely related CR-NA-C1-C2

dihedral, MM give a small, negative V1coefficient, ABS set this to zero (AMBER values for generic X-N*-CT-X), and MSB keep the same negative V1and V2coefficients as for CW- NA-C1-C2. These two functions make up a very important contribution to the torsion energy of the cation. They are correlated by an angle of 180°because of the ring geometry, because CRand CW are both connected to the nitrogen atom.

So these two functions were fitted simultaneously by us, and Figure 3. Torsion profile of the dihedrals involving imidazolium ring to aliphatic side chain bonds in 1-alkyl-3-methylimidazolium cations.

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the values found for the parameters of both dihedral functions are mutually different, as can be seen in Table 3.

Finally, for dihedral NA-C1-C2-CT, MM set it to zero, ABS use AMBER values for generic X-CT-CT-X, and MSB adopt coefficients much larger than ours, taken from OPLS-AA for aliphatic amines. In the case of NA-C1-C2-HC we obtained negligible coefficients, MM also set this term to zero, and ABS use AMBER values for generic X-CT-CT-X, whereas MSB adopt again large coefficients, taken from OPLS-AA for aliphatic amines.

In conclusion to this section, we justify the effort undertaken here, for the first time to our knowledge, to calculate in a rigorous manner the dihedral terms affecting the junction between the aromatic ring and the alkyl side chain for the family of imidazolium cations. The force fields proposed in the literature so far contain some crude approximations and guesswork in this particular aspect, which certainly affect the conformational details of the ionic liquids.

Ab Initio Calculation of Partial Charges. Atomic charges on the imidazolium ring and its adjacent atoms were also obtained in the present study. They were calculated for mmim⁺, emim⁺, and bmim⁺by electrostatic surface potential fits, using the CHelpG procedure,²⁰to electron densities obtained at the MP2/cc-pVTZ(-f) level. For emim⁺and bmim⁺ the charges correspond to the average of two different geometries at conformational energy minima. The results are shown in Figure 4. To ensure the transferability of the model along the entire dialkylimidazolium cation family: (i) The charges on the atoms of the ring, those directly attached to it and the H1hydrogen atoms, were given symmetrical values along the C2Vaxis of the ring and have values close to those found for mmim⁺. For bmim⁺and emim⁺the differences, including departures from symmetry, are small; cf. Figure 4. (ii) The charges on all atoms in the alkyl side chain removed from the ring more than 3 bonds (CS or CT and HC) were given the corresponding OPLS-AA values for alkanes;⁸(iii) The charge in the remaining atom that establishes the connection between the ring and the side chain, CEin emim⁺and C2in longer side chains, was found empirically to respect the total charge of the cation. The values of the

adopted charges are listed in Table 3 and compared with results from other force fields in Table 5.

Although we performed calculations on a larger basis set and employed a method including electron correlation (MP2), the charges obtained in this work show an overall agreement with those proposed by several other authors that used electrostatic surface potential fits to the ab initio electron densities. An exception is the relative charge distribution between the nitrogen atoms and carbon atom between them: our NA atoms are somewhat more positive and this is compensated by a less positive CR atom. The distributed multipole analysis method employed by Hanke et al.²yielded a significantly different set of atomic charges, in both sign and magnitude, when compared with all other authors.

MD Simulations. The validation of the proposed force field was based on simulation results obtained using the molecular dynamics technique, implemented with the DL_POLY software.¹⁹ The strategy was simple: because the most accurate and common experimental data concerning ILs are densities, we decided to test the performance of the proposed force field in the estimation of molar densities along the family of the imidazolium cation, with different counterions, in both the crystalline and liquid phases.

Crystalline Phase. Five crystalline structures containing the imidazolium cation were selected from the Cambridge Crystal- lographic Database (CCDB). These were the sole mmim⁺-based structure found in the database ([mmim][Cl]), three of the more than thirty emim⁺-based structures ([emim][Cl], [emim][NO3], and [emim][PF6]), and a ddmim⁺structure ([ddmim][PF6]). For 1-alkyl-3-methylimidazolium cations with alkyl side chains ranging from three to eleven carbon atoms there are very few crystallographic structures recorded at the CCDB, all of them including large or heavy counterions. This fact reflects the low melting point temperatures of imidazolium ILs with intermediate side-chain length; see Discussion section.

The simulation boxes and initial configurations were set by taking into account the dimensions and occupancy of the unit cells of each crystalline structure. Because the dimensions of the unit cells of these crystals are too small to accommodate a Figure 4. Atomic partial charges of the imidazolium ring and adjacent atoms. All numbers are percentage values of the charge of one proton. The other carbon and hydrogen atoms in the alkyl side chain (not shown) were given OPLS-AA charges. The list at each atom corresponds to the charge values obtained from ab initio calculations for mmim⁺, emim⁺, and bmim⁺and (in bold) the value adopted in the present force field, respectively.

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sufficiently large cutoff distance, several cells were stacked together to form a sufficiently large and well proportioned simulation box, with cutoff distances ranging 12.5-16 Å. Long- range corrections were applied beyond the cutoff distance and the Ewald method was implemented to take into account the long-range character of the electrostatic interactions. The initial position, orientation and conformation of each molecule within the simulation box was the one defined by the lattice coordinates at the CCDB. Because the overall size of the simulation box is defined by the dimensions of the unit cell of each crystal, simulations with different box sizes and cutoff distances were run to check that the dimensions of the simulation box/cutoff were sufficiently large to make negligible any finite size effects.

The simulation details for each crystal are given in Table 6.

The simulations were performed using a Nose´-Hoover thermostat coupled with an anisotropic Hoover barostat that allowed the simulation box to change volume and shape under N-p-T conditions. The temperature was fixed to match those used during the crystallographic experiments, and the pressure was set to a null value. All runs were allowed to equilibrate for a period of 100 ps, followed by production times of 200 ps.

These simulation times were found appropriate because we start from known configurations close to the equilibrium structure, and it is observed that the relaxation is complete before the end of the equilibration period.

The objective of these simulations is not to test the ability of the present force field to generate the corresponding experimental crystal lattice, a procedure that is still controversial even for simple molecular crystals, but simply to check if they are compatible. The stringency of the test was confirmed with

simulation runs where ad hoc parameters were introduced and large distortions of the unit cell paramenters of the lattice observed. The use of an anisotropic barostat coupled with the system allows for a relatively short equilibration period due to the frequent rescaling of the position of the particles. The rather short but effective relaxation time was also confirmed by monitoring the length of the sides and angles of the simulation box as the simulation proceeded through the equilibration period.

Liquid Phase. The ionic liquids simulated were the combinations of emim⁺, bmim⁺, and hmim⁺with PF6-

, NO3-

, and Cl^-. For ionic liquids containing hmim⁺, simulations were performed in periodic cubic boxes containing 200 pairs of ions, whereas for emim⁺and bmim⁺, 250 pairs were considered. Spherical cutoff distances were defined at 11, 12, and 13 Å for the emim⁺, bmim⁺, and hmim⁺salts, respectively. Tail corrections to short- range interactions were included, and the long-range electrostatic term was dealt with by the Ewald summation method. The equilibration period in the liquid phase simulation runs is very important. Initially, the ions are placed at random in the simulation box, and the equilibration starts by a short relaxation of a few picoseconds at 1 K and constant N-V-E. Then an equilibration period is imposed at the final temperature of the simulation, followed by the activation of the thermostat and barostat. Production runs were started after equilibrations of at least 200 ps.

Production runs of 300 ps were made in N-p-T conditions, under a pressure of 1 bar, and at 298 K for all simulations except for [bmim][NO3], where the system was simulated at 313 K to compare the results with experimental values at the same TABLE 5: Charge Distribution in 1,3-Alkylmethylimidazolium Cations According to Different Force Fields^a

model (authors)

HPLB [ref 2] ABS [refs 3,5] SBM [ref 4] MM [ref 7] MSB [ref 6] CLDP this work Ions Modeled

cations mmim⁺, emim⁺ emim⁺bmim⁺ Bmim⁺ bmim⁺ bmim⁺ amim⁺(C1-C12)

Method (Based on ab Initio Calculations)

AI level of theory MP2 UHF RHF B3LYP HF MP2

AI basis set 6-31G(d,p) 6-31G(d) 6-31G(d) 6-311+G(d) 6-31G(d,p) cc-pVTZ(-f)

partial charges DMA RESP CHelpG CHelpG ESP CHelpG

mâ av aâ mâ av a^b m^b av a^b m^b av a^b

Ring Atoms

NA -27 7 4 2 13 10 7 13 12 11 24 15 7 15

CR 41 3 6 -6 -11

(CR+HA) 50 25 23 23 18 10

HA 10 22 18 24 21

CW 11 -15 -18 -21 -14 -18 -22 -28 -20 -12 -13

(CW+HA) 20 9 7 4 4 6 10 4 2 -1 -2 5 12 7

HA 9 24 25 26 18 19 21 26 24 24 21

Side-Chain Atoms

C1 12 -16 -9 -2 -16 -4 9 -33 -25 -17 -17

(C1+3H1) 31 23 24 25 22 17 19 24 17 22 26 22

H1 6 13 11 9 11 8 5 17 16 14 13

CE -6 -8 -5

(CE+3HC) 8 9 13

HC 5 6 6

C2 3 -12 1

(C2+2HC) 7 12 -7 no data, OLPS ? 13

HC 2 3 6

CS 5 26 -12

(CS+2HC) 8 12 13 no data, OLPS ? 0

HC 2 -6 6

CT -16 -21 0 -18

(CT+3HC) - 1 -5 -4 12 0

HC 5 6 4 6

aAll numbers are percentage values of the charge of one proton.^bThe m and a labels refer to atoms on the side of the methyl and alkyl side chains, repectively.